#501217

order	512
nilpotency class	4
isoclinism	isoclinic class of #499712
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₄	D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1092 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2,3,4)(9,10)(11,21)(12,24)(13,27)(14,17)(15,33)(16,22)(18,25)(19,32)(20,30)(23,38)(26,37)(28,36)(29,34)(31,39)(35,40);(5,6)(7,8)(9,11,14,12)(10,16,17,18)(13,28,31,29)(15,26,30,23)(19,37,35,38)(20,36,33,34)(21,39,24,27)(22,40,25,32);(2,4)(6,8)(10,17)(16,18)(19,35)(20,33)(21,24)(27,39)(34,36)(37,38);(9,12,14,11)(10,18,17,16)(13,29,31,28)(15,23,30,26)(19,38,35,37)(20,34,33,36)(21,27,24,39)(22,32,25,40)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,d^2=i,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,e^b=eh,h^b=hi,e^d=eh,h^d=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408413);